Illustration of SuSiE in a “difficult” situation

Last updated: 2020-01-15

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Rmd	730b73b	Peter Carbonetto	2020-01-15	wflow_publish(“susie_jrssb_example.Rmd”)
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html	72fe4c0	Peter Carbonetto	2020-01-15	I have a complete first draft of the susie example.
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html	81df87b	Peter Carbonetto	2020-01-15	Added more to susie example.
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Rmd	a70ba28	Peter Carbonetto	2020-01-14	Removed old home page.
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html	f73d1cb	Peter Carbonetto	2020-01-14	First build of the susie example.
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Here we explore the behaviour of SuSiE in a toy example where the correlations between the predictors are not so straightforward. Despite being more complicated, SuSiE still correctly infers Credible Sets.

Load packages

The mvtnorm package is used to simulate the data.

library(mvtnorm)
library(varbvs)
library(susieR)

Overview

One of the referees was interested in understanding the behaviour of SuSiE in an example in which the correlation structure is slightly more complex than the motivating example provided in the paper.

Consider five correlated predictors in a simple linear regression, $y = x_1\beta_1, \ldots, x_5\beta_5$, in which only two variables, $x1$ and $x3$, have an effect on the outcome, $y$; that is, the coefficients $\beta_1$ and $\beta_3$ are both nonzero. The first two predictors are very strongly correlated with each other, and the third and fourth predictors are also very strongly correlated. Therefore, it will be difficult to distinguish between $x1$ and $x2$, and likewise for $x3$ and $x4$. This correlation structure is the same as the simple motivating example given in the paper.

To complicate the example, there is a fifth predictor that has no effect on the outcome, and is also strongly correlated with the other variables. This is the correlation matrix for the five predictors:

S <- rbind(c(   1, 0.99,  0.8,  0.8, 0.9),
           c(0.99,    1,  0.8,  0.8, 0.9),
           c( 0.8,  0.8,    1, 0.99, 0.9),
           c( 0.8,  0.8, 0.99,    1, 0.9),
           c( 0.9,  0.9,  0.9,  0.9,   1))

Therefore, without an abundance of data, it will be difficult to distinguish between variables 1, 2 and 5, and likewise for variables 3, 4 and 5. The inferential statement that would best capture our uncertainty about which variables affect the outcome would look something like this:

\[\mbox{Among coefficients $\beta_1, \ldots, \beta_5$, two (or more) are not zero}\]

SuSiE is not designed to produce such inferential statements, but could state instead:

\[(\beta_1 \neq 0 \mbox{ or } \beta_2 \neq 0 \mbox{ or } \beta_5 \neq 0) \mbox{ and } (\beta_3 \neq 0 \mbox{ or } \beta_4 \neq 0)\]

Another valid statement could be:

\[(\beta_1 \neq 0 \mbox{ or } \beta_2 \neq 0) \mbox{ and } (\beta_3 \neq 0 \mbox{ or } \beta_4 \neq 0 \mbox{ or } \beta_5 \neq 0)\]

Although neither of these statements completely capture our uncertainty about which variables affect the outcome, it does satisfy our aim as it is stated in the paper:

"A key feature of our method, which distinguishes it from most existing BVSR methods, is that it produces Credible Sets of variables that quantify uncertainty in which variable should be selected when multiple, highly correlated variables compete with one another. These Credible Sets are designed to be as small as possible while still each capturing a relevant variable.

The key point here is that the Credible Sets are not intended to capture full uncertainty in which variables should be selected. But they are intended to capture all relevant variables.

Below, we investigate these claims empirically.

Simulate data

We simulate 200 samples in this scenario.

set.seed(1)
n <- 200
b <- c(1, 0, -1, 0, 0)
X <- rmvnorm(n,sigma = S)
y <- drop(X %*% b + rnorm(n)/0.3)

Multiple regression using varbvs

First, to provide a counterexample that does not accomplish what we set out to achieve, we fit the “varbvs” model, and show that varbvs, although similar to SuSiE in many ways, does not adequately capture uncertainty in the selected variables.

fit1 <- varbvs(X,NULL,y,logodds = log10(0.5),sa = 1,verbose = FALSE)
summary(fit1)$top.vars
#   index variable       prob         PVE          coef  Pr(coef.>0.95)
# 1     3       X3 0.99995825 0.128995948 -1.3311483474 [-1.836,-0.826]
# 2     1       X1 0.98264586 0.073088751  0.9266196082 [+0.460,+1.425]
# 3     4       X4 0.03756227 0.004674475  0.0005357177 [-0.495,+0.524]
# 4     5       X5 0.03643073 0.004849222 -0.0017998663 [-0.534,+0.435]
# 5     2       X2 0.03618984 0.004751830  0.0012398904 [-0.452,+0.521]

varbvs has correctly determined that two variables are useful for predicting the outcome. However, it incorrectly placed all the weight on the first and third variables. This is expected behaviour—it is due to the particular variational approximation used in varbvs—but it is undesirable when we want to better quantify uncertainty in the correlated variables.

Multiple regression using SuSiE

Next, we fit a SuSiE model. (For simplicity, we fit a SuSiE model with $L = 2$, but SuSiE will give similar results for $L > 2$.)

fit2 <- susie(X,y,L = 2,standardize = FALSE,estimate_prior_variance = FALSE,
              scaled_prior_variance = 1,min_abs_corr = 0)
susie_get_cs(fit2,X,min_abs_corr = 0)$cs
# $L2
# [1] 1 2 5
# 
# $L1
# [1] 3 4 5

SuSiE identified two Credible Sets:

\[(\beta_1 \neq 0 \mbox{ or } \beta_2 \neq 0 \mbox{ or } \beta_5 \neq 0) \mbox{ and } (\beta_3 \neq 0 \mbox{ or } \beta_4 \neq 0 \mbox{ or } \beta_5 \neq 0)\]

Looking more closely at the posterior inclusion probabilities, most of the weight is assigned to $x_3$ and $x_4$ in the first Credible Set, and to $x_1$ and $x_2$ in the second Credible Set, and, in light of the strong correlations with $x_5$, each Credible Set also allows for the possibility that $x_5$ affects the outcome:

round(fit2$alpha,digits = 3)
#       [,1]  [,2]  [,3]  [,4]  [,5]
# [1,] 0.001 0.001 0.628 0.319 0.051
# [2,] 0.466 0.477 0.010 0.012 0.035

Therefore, although this statement is not the same as, “among coefficients $\beta_1, \ldots, \beta_5$, two (or more) are not zero”, SuSiE still has correctly achieves the useful aim of identifying “Credible Sets” that capture all relevant variables.